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Coordinative structures as scale-free networks: Cascade and percolation dynamics in motor learning with empirical validation.

Code ↔ Paper

14 matches between paragraphs of the paper and lines of its authors' code, computed by the harvester (lexical-v1). Click a colored paragraph or line to see its counterpart.

The 14 matches
  1. [1] § 2. Materials and methods › 2.2. Cascade dynamics model ↔ simulations/02_cascade_dynamics.py, lines 115–208 · score 0.90 · neighbor mediated propagation, state variables, spontaneous activation, state transition, coordination strength, timestep
  2. [2] § 3. Results › 3.2. Cascade dynamics ↔ simulations/02_cascade_dynamics.py, lines 1–44 · score 0.86 · topology isolated trials, DOF recruitment, spontaneous activation, single seed, asymmetry ratio, Cascade dynamics
  3. [3] § 2. Materials and methods › 2.4. Percolation model and coupled learning dynamics ↔ simulations/04_coupled_learning.py, lines 1–41 · score 0.82 · Hebbian weight updates, preferential attachment, step coupled, practice iteration, fitness, 2–4
  4. [4] § 2. Materials and methods › 2.1. Network model specifications ↔ simulations/01_network_generation.py, lines 1–42 · score 0.78 · Barab si Albert, canonical network, Watts Strogatz, scale free networks, Erd, nyi
  5. [5] § 2. Materials and methods › 2.2. Cascade dynamics model ↔ simulations/02_cascade_dynamics.py, lines 1–44 · score 0.74 · baseline stability, centrality weighted, coordination strength, neighbors, S2, activation
  6. [6] § 3. Results › 3.4. Coupled learning dynamics ↔ simulations/04_coupled_learning.py, lines 1–41 · score 0.72 · Hebbian learning rate, coupled learning dynamics, practice iterations, weight hierarchy, algorithm, decay
  7. [7] § 2. Materials and methods › 2.1. Network model specifications ↔ simulations/01_network_generation.py, lines 1–42 · score 0.65 · small world network, power law, random network, scale free network, clustering, connecting
  8. [8] § 2. Materials and methods › 2.4. Percolation model and coupled learning dynamics ↔ simulations/03_percolation_robustness.py, lines 82–127 · score 0.58 · giant component fraction, occupation probability, bond, percolation
  9. [9] § 3. Results › 3.1. Network topology comparison ↔ simulations/01_network_generation.py, lines 244–371 · score 0.57 · exemplar network, hub fraction, eigenvector centrality, Gini, scale free network, seed
  10. [10] § 3. Results › 3.3. Percolation dynamics and robustness-fragility ↔ simulations/03_percolation_robustness.py, lines 1–41 · score 0.56 · bond occupation sweeps, robustness, susceptibility, fragility, percolation, realizations
  11. [11] § 2. Materials and methods › 2.3. Mapping network topology to coordinative structures ↔ simulations/03_percolation_robustness.py, lines 173–196 · score 0.56 · random node removal, fragmentation ratio, Robustness
  12. [12] § 3. Results › 3.1. Network topology comparison ↔ simulations/01_network_generation.py, lines 103–124 · score 0.54 · Gini coefficient, hub fraction, eigenvector centrality, nodes, networks
  13. [13] § 2. Materials and methods › 2.4. Percolation model and coupled learning dynamics ↔ simulations/04_coupled_learning.py, lines 147–176 · score 0.54 · co activated, coupled learning, decay, strengthen, Edges, weights
  14. [14] § 3. Results › 3.1. Network topology comparison ↔ simulations/01_network_generation.py, lines 244–371 · score 0.53 · Power law fitting, network ensembles, KS, WS, BA, topologies

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The authors' code

Python · 455 lines · 17 KB · MIT · 5 matches

  1. #!/usr/bin/env python3
  2. """
  3. 01_network_generation.py
  4. ========================
  5. Coordinative Structures as Scale-Free Networks (Park, 2026)
  6. Manuscript Section 2.1 — Network Topology
  8. Generates three canonical network topologies with matched connectivity:
  9. - Erdős–Rényi (ER) random network
  10. - Watts–Strogatz (WS) small-world network
  11. - Barabási–Albert (BA) scale-free network
  13. Computes structural metrics for manuscript Tables S1a–S1d and Table 2:
  14. - Degree statistics: ⟨k⟩, σ_k, k_max
  15. - Degree heterogeneity: κ = ⟨k²⟩/⟨k⟩
  16. - Power-law exponent: γ̂ (MLE) with KS goodness-of-fit
  17. - Centrality: eigenvector centrality, Gini coefficient, hub fraction
  18. - Topology: clustering coefficient, average path length
  20. Parameters (from manuscript §2.1):
  21. N = 100 nodes
  22. ⟨k⟩ ≈ 6 (matched across topologies)
  23. ER: p = 0.0606
  24. WS: K = 6, β = 0.1
  25. BA: m = 3 (m₀ = 3 initial complete graph)
  26. Realizations: 100
  27. Base random seed: 42
  29. Output:
  30. data/simulation_outputs/network_structural_metrics.csv
  31. data/simulation_outputs/network_ensemble_data.npz
  33. Usage:
  34. python simulations/01_network_generation.py
  35. """
  37. import numpy as np
  38. import networkx as nx
  39. from scipy import stats
  40. import os
  41. import sys
  42. import time
  44. # ══════════════════════════════════════════════════════════════
  45. # PARAMETERS — manuscript §2.1, Tables S1b, S1d
  46. # ══════════════════════════════════════════════════════════════
  47. N = 100 # Number of nodes (degrees of freedom)
  48. N_REALIZATIONS = 100 # Independent network instances
  49. BASE_SEED = 42 # For reproducibility
  51. # Topology-specific parameters (Table S1b)
  52. ER_P = 6.0 / (N - 1) # ≈ 0.0606, targeting ⟨k⟩ ≈ 6
  53. WS_K = 6 # Ring neighbors (each side K/2 = 3)
  54. WS_BETA = 0.1 # Rewiring probability
  55. BA_M = 3 # Edges per new node (⟨k⟩ → 2m = 6)
  57. # Hub identification threshold
  58. HUB_CRITERION = 'mean_plus_sd' # C_e > μ + σ
  60. # Output directory
  61. OUTPUT_DIR = os.path.join(os.path.dirname(os.path.dirname(
  62. os.path.abspath(__file__))), 'data', 'simulation_outputs')
  65. # ══════════════════════════════════════════════════════════════
  66. # NETWORK GENERATION FUNCTIONS
  67. # ══════════════════════════════════════════════════════════════
  68. def generate_er(n, p, seed):
  69. """Erdős–Rényi G(N,p) random network (Eq S1 Appendix, Algorithm 1)."""
  70. return nx.erdos_renyi_graph(n, p, seed=seed)
  73. def generate_ws(n, k, beta, seed):
  74. """Watts–Strogatz small-world network (Algorithm 2)."""
  75. return nx.watts_strogatz_graph(n, k, beta, seed=seed)
  78. def generate_ba(n, m, seed):
  79. """Barabási–Albert scale-free network via preferential attachment (Eq 1)."""
  80. return nx.barabasi_albert_graph(n, m, seed=seed)
  83. # ══════════════════════════════════════════════════════════════
  84. # STRUCTURAL ANALYSIS FUNCTIONS
  85. # ══════════════════════════════════════════════════════════════
  86. def compute_degree_stats(G):
  87. """Compute degree distribution statistics."""
  88. degrees = np.array([G.degree(n) for n in G.nodes()])
  89. k_mean = degrees.mean()
  90. k_std = degrees.std()
  91. k_max = degrees.max()
  92. k_second_moment = (degrees ** 2).mean()
  93. kappa = k_second_moment / k_mean if k_mean > 0 else 0.0
  94. return {
  95. 'k_mean': k_mean,
  96. 'k_std': k_std,
  97. 'k_max': k_max,
  98. 'kappa': kappa,
  99. 'degrees': degrees,
  100. }
  103. def compute_centrality_stats(G):
  104. """Compute eigenvector centrality and derived measures."""
  105. try:
  106. ce = nx.eigenvector_centrality_numpy(G)
  107. except nx.NetworkXError:
  108. # Fallback for disconnected graphs
  109. ce = nx.eigenvector_centrality(G, max_iter=1000, tol=1e-6)
  110. ce_vals = np.array([ce[n] for n in G.nodes()])
  112. # Gini coefficient of eigenvector centrality
  113. gini = _gini_coefficient(ce_vals)
  115. # Hub fraction: nodes with C_e > μ + σ
  116. ce_mean = ce_vals.mean()
  117. ce_std = ce_vals.std()
  118. hub_fraction = np.mean(ce_vals > (ce_mean + ce_std))
  120. return {
  121. 'gini_ce': gini,
  122. 'hub_fraction': hub_fraction,
  123. 'ce_values': ce_vals,
  124. }
  127. def _gini_coefficient(values):
  128. """Compute Gini coefficient for a set of values."""
  129. sorted_vals = np.sort(values)
  130. n = len(sorted_vals)
  131. if n == 0 or sorted_vals.sum() == 0:
  132. return 0.0
  133. index = np.arange(1, n + 1)
  134. return (2 * np.sum(index * sorted_vals) / (n * np.sum(sorted_vals))) - (n + 1) / n
  137. def compute_topology_stats(G):
  138. """Compute clustering coefficient and average path length."""
  139. clustering = nx.average_clustering(G)
  141. # Average shortest path length (handle disconnected graphs)
  142. if nx.is_connected(G):
  143. avg_path = nx.average_shortest_path_length(G)
  144. else:
  145. # Use largest connected component
  146. gcc = max(nx.connected_components(G), key=len)
  147. subG = G.subgraph(gcc).copy()
  148. avg_path = nx.average_shortest_path_length(subG) if len(subG) > 1 else float('inf')
  150. return {
  151. 'clustering': clustering,
  152. 'avg_path_length': avg_path,
  153. }
  156. def powerlaw_fit(degrees, k_min=None):
  157. """
  158. Maximum Likelihood Estimation of power-law exponent γ̂.
  160. Uses the discrete MLE formula (Eq S1 in S1 Appendix):
  161. γ̂ = 1 + n [Σᵢ ln(kᵢ / (k_min − 0.5))]⁻¹
  163. Following Clauset, Shalizi & Newman (2009) [Ref 58 in manuscript].
  165. Returns:
  166. gamma_hat: MLE exponent estimate
  167. ks_pvalue: KS test p-value against power-law distribution
  168. k_min_used: k_min threshold used
  169. """
  170. degrees = np.array(degrees)
  171. degrees = degrees[degrees > 0] # Remove zeros
  173. if k_min is None:
  174. # Find optimal k_min by minimizing KS statistic
  175. unique_k = np.unique(degrees)
  176. unique_k = unique_k[unique_k >= 2] # Minimum meaningful k_min
  177. best_ks = np.inf
  178. best_kmin = 2
  179. best_gamma = 3.0
  181. for km in unique_k:
  182. if km >= degrees.max():
  183. break
  184. tail = degrees[degrees >= km]
  185. if len(tail) < 10:
  186. break
  187. g = 1 + len(tail) * (np.sum(np.log(tail / (km - 0.5)))) ** (-1)
  188. if g > 1.5 and g < 5.0: # Reasonable range
  189. # KS statistic
  190. cdf_emp = np.sort(tail)
  191. cdf_emp = np.arange(1, len(cdf_emp) + 1) / len(cdf_emp)
  192. cdf_theo = 1 - (np.sort(tail) / km) ** (-(g - 1))
  193. ks = np.max(np.abs(cdf_emp - cdf_theo))
  194. if ks < best_ks:
  195. best_ks = ks
  196. best_kmin = km
  197. best_gamma = g
  199. k_min = best_kmin
  201. # Final fit with chosen k_min
  202. tail = degrees[degrees >= k_min]
  203. if len(tail) < 5:
  204. return np.nan, np.nan, k_min
  206. gamma_hat = 1 + len(tail) * (np.sum(np.log(tail / (k_min - 0.5)))) ** (-1)
  208. # KS goodness-of-fit via Monte Carlo (simplified)
  209. n_mc = 500
  210. ks_orig = _ks_stat_powerlaw(tail, gamma_hat, k_min)
  211. n_exceed = 0
  212. rng = np.random.RandomState(BASE_SEED)
  213. for _ in range(n_mc):
  214. # Generate synthetic power-law sample
  215. synth = _generate_powerlaw_sample(len(tail), gamma_hat, k_min, rng)
  216. if len(synth) < 5:
  217. continue
  218. g_synth = 1 + len(synth) * (np.sum(np.log(synth / (k_min - 0.5)))) ** (-1)
  219. ks_synth = _ks_stat_powerlaw(synth, g_synth, k_min)
  220. if ks_synth >= ks_orig:
  221. n_exceed += 1
  223. ks_pvalue = n_exceed / n_mc
  225. return gamma_hat, ks_pvalue, k_min
  228. def _ks_stat_powerlaw(data, gamma, k_min):
  229. """KS statistic between data and power-law CDF."""
  230. sorted_data = np.sort(data)
  231. n = len(sorted_data)
  232. cdf_emp = np.arange(1, n + 1) / n
  233. cdf_theo = 1 - (sorted_data / k_min) ** (-(gamma - 1))
  234. return np.max(np.abs(cdf_emp - cdf_theo))
  237. def _generate_powerlaw_sample(n, gamma, k_min, rng):
  238. """Generate discrete power-law random sample via inverse transform."""
  239. u = rng.uniform(0, 1, size=n)
  240. samples = np.floor(k_min * (1 - u) ** (-1.0 / (gamma - 1))).astype(int)
  241. return samples[samples >= k_min]
  244. # ══════════════════════════════════════════════════════════════
  245. # MAIN ENSEMBLE SIMULATION
  246. # ══════════════════════════════════════════════════════════════
  247. def run_ensemble():
  248. """
  249. Generate network ensembles and compute all structural metrics.
  250. Reproduces Tables S1b, S1d, and Table 2 (structural rows).
  251. """
  252. print("=" * 70)
  253. print("01_network_generation.py")
  254. print("Coordinative Structures as Scale-Free Networks")
  255. print(f"N = {N}, ⟨k⟩ ≈ 6, {N_REALIZATIONS} realizations, seed = {BASE_SEED}")
  256. print("=" * 70)
  258. topologies = {
  259. 'ER': {'generator': lambda seed: generate_er(N, ER_P, seed),
  260. 'params': f'p = {ER_P:.4f}'},
  261. 'WS': {'generator': lambda seed: generate_ws(N, WS_K, WS_BETA, seed),
  262. 'params': f'K = {WS_K}, β = {WS_BETA}'},
  263. 'BA': {'generator': lambda seed: generate_ba(N, BA_M, seed),
  264. 'params': f'm = {BA_M}'},
  265. }
  267. # Storage for ensemble results
  268. results = {}
  269. ensemble_data = {}
  271. for topo_name, topo_info in topologies.items():
  272. print(f"\n--- {topo_name} ({topo_info['params']}) ---")
  273. t0 = time.time()
  275. # Per-realization storage
  276. k_means = []
  277. k_stds = []
  278. k_maxs = []
  279. kappas = []
  280. clusterings = []
  281. path_lengths = []
  282. gini_ces = []
  283. hub_fracs = []
  284. gamma_hats = []
  285. ks_pvals = []
  286. all_degrees = []
  288. for r in range(N_REALIZATIONS):
  289. seed = BASE_SEED + r
  290. G = topo_info['generator'](seed)
  292. # Degree statistics
  293. dstats = compute_degree_stats(G)
  294. k_means.append(dstats['k_mean'])
  295. k_stds.append(dstats['k_std'])
  296. k_maxs.append(dstats['k_max'])
  297. kappas.append(dstats['kappa'])
  298. all_degrees.append(dstats['degrees'])
  300. # Centrality statistics
  301. cstats = compute_centrality_stats(G)
  302. gini_ces.append(cstats['gini_ce'])
  303. hub_fracs.append(cstats['hub_fraction'])
  305. # Topology statistics
  306. tstats = compute_topology_stats(G)
  307. clusterings.append(tstats['clustering'])
  308. path_lengths.append(tstats['avg_path_length'])
  310. # Power-law fit (BA only — ER and WS are not power-law)
  311. if topo_name == 'BA':
  312. gamma, ks_p, _ = powerlaw_fit(dstats['degrees'])
  313. gamma_hats.append(gamma)
  314. ks_pvals.append(ks_p)
  316. elapsed = time.time() - t0
  317. print(f" Completed {N_REALIZATIONS} realizations in {elapsed:.1f}s")
  319. # Ensemble statistics
  320. res = {
  321. 'k_mean': (np.mean(k_means), np.std(k_means)),
  322. 'k_std': (np.mean(k_stds), np.std(k_stds)),
  323. 'k_max': (np.mean(k_maxs), np.std(k_maxs)),
  324. 'kappa': (np.mean(kappas), np.std(kappas)),
  325. 'clustering': (np.mean(clusterings), np.std(clusterings)),
  326. 'avg_path': (np.mean(path_lengths), np.std(path_lengths)),
  327. 'gini_ce': (np.mean(gini_ces), np.std(gini_ces)),
  328. 'hub_fraction': (np.mean(hub_fracs), np.std(hub_fracs)),
  329. }
  331. if topo_name == 'BA':
  332. valid_gamma = [g for g in gamma_hats if not np.isnan(g)]
  333. valid_ks = [p for p in ks_pvals if not np.isnan(p)]
  334. res['gamma_hat'] = (np.mean(valid_gamma), np.std(valid_gamma))
  335. res['ks_pvalue'] = (np.mean(valid_ks), np.std(valid_ks))
  337. results[topo_name] = res
  339. # Store ensemble arrays for downstream use
  340. ensemble_data[f'{topo_name}_kappas'] = np.array(kappas)
  341. ensemble_data[f'{topo_name}_gini'] = np.array(gini_ces)
  342. ensemble_data[f'{topo_name}_hub_frac'] = np.array(hub_fracs)
  343. ensemble_data[f'{topo_name}_k_means'] = np.array(k_means)
  344. ensemble_data[f'{topo_name}_k_stds'] = np.array(k_stds)
  345. ensemble_data[f'{topo_name}_k_maxs'] = np.array(k_maxs)
  346. ensemble_data[f'{topo_name}_clusterings'] = np.array(clusterings)
  347. ensemble_data[f'{topo_name}_path_lengths'] = np.array(path_lengths)
  349. # Store exemplar network (seed=42) for visualization
  350. G_exemplar = topo_info['generator'](BASE_SEED)
  351. pos_exemplar = nx.spring_layout(G_exemplar, seed=BASE_SEED,
  352. k=1.8 / np.sqrt(N), iterations=80)
  353. deg_exemplar = np.array([G_exemplar.degree(n) for n in G_exemplar.nodes()])
  354. try:
  355. ce_exemplar = np.array(list(
  356. nx.eigenvector_centrality_numpy(G_exemplar).values()))
  357. except Exception:
  358. ce_exemplar = np.zeros(N)
  360. ce_norm = (ce_exemplar - ce_exemplar.min()) / (
  361. ce_exemplar.max() - ce_exemplar.min() + 1e-10)
  363. pos_array = np.array([pos_exemplar[n] for n in G_exemplar.nodes()])
  364. edges_array = np.array(list(G_exemplar.edges()))
  366. ensemble_data[f'net_{topo_name}_pos'] = pos_array
  367. ensemble_data[f'net_{topo_name}_degs'] = deg_exemplar
  368. ensemble_data[f'net_{topo_name}_ce'] = ce_norm
  369. ensemble_data[f'net_{topo_name}_edges'] = edges_array
  371. return results, ensemble_data
  374. def print_results(results):
  375. """Print results in Table S1d format."""
  376. print("\n" + "=" * 70)
  377. print("TABLE S1d — Structural validation measures")
  378. print(f"(ensemble means ± SD; {N_REALIZATIONS} realizations; "
  379. f"N = {N}, ⟨k⟩ ≈ 6, seed base = {BASE_SEED})")
  380. print("=" * 70)
  382. measures = [
  383. ('⟨k⟩', 'k_mean'),
  384. ('σ_k', 'k_std'),
  385. ('k_max', 'k_max'),
  386. ('κ = ⟨k²⟩/⟨k⟩', 'kappa'),
  387. ('Clustering', 'clustering'),
  388. ('Avg path', 'avg_path'),
  389. ('Gini(C_e)', 'gini_ce'),
  390. ('Hub fraction', 'hub_fraction'),
  391. ]
  393. header = f"{'Measure':<18} {'ER (Random)':<20} {'WS (Small-World)':<20} {'BA (Scale-Free)':<20}"
  394. print(header)
  395. print("-" * 78)
  397. for label, key in measures:
  398. row = f"{label:<18}"
  399. for topo in ['ER', 'WS', 'BA']:
  400. mean, sd = results[topo][key]
  401. row += f" {mean:>7.2f} ± {sd:<6.2f} "
  402. print(row)
  404. # BA-specific measures
  405. if 'gamma_hat' in results['BA']:
  406. mean, sd = results['BA']['gamma_hat']
  407. print(f"{'γ̂ (MLE)':<18} {'---':<20} {'---':<20} {mean:>7.2f} ± {sd:<6.2f}")
  408. if 'ks_pvalue' in results['BA']:
  409. mean, sd = results['BA']['ks_pvalue']
  410. print(f"{'KS p-value':<18} {'---':<20} {'---':<20} {mean:>7.2f} ± {sd:<6.2f}")
  412. # Manuscript Table 2 comparison
  413. print("\n" + "=" * 70)
  414. print("VERIFICATION against manuscript Table 2")
  415. print("=" * 70)
  416. expected = {
  417. 'ER': {'kappa': 6.84, 'gini_ce': 0.26, 'hub_fraction': 0.16},
  418. 'WS': {'kappa': 6.09, 'gini_ce': 0.16, 'hub_fraction': 0.16},
  419. 'BA': {'kappa': 9.67, 'gini_ce': 0.35, 'hub_fraction': 0.10},
  420. }
  421. for topo in ['ER', 'WS', 'BA']:
  422. print(f"\n {topo}:")
  423. for key, exp_val in expected[topo].items():
  424. sim_val = results[topo][key][0]
  425. match = "✓" if abs(sim_val - exp_val) / (exp_val + 1e-10) < 0.15 else "✗"
  426. print(f" {key:<15} expected={exp_val:.2f} simulated={sim_val:.2f} {match}")
  429. def save_results(results, ensemble_data):
  430. """Save results to CSV and NPZ files."""
  431. os.makedirs(OUTPUT_DIR, exist_ok=True)
  433. # CSV summary
  434. csv_path = os.path.join(OUTPUT_DIR, 'network_structural_metrics.csv')
  435. with open(csv_path, 'w') as f:
  436. f.write("topology,measure,mean,sd\n")
  437. for topo in ['ER', 'WS', 'BA']:
  438. for key, (mean, sd) in results[topo].items():
  439. f.write(f"{topo},{key},{mean:.6f},{sd:.6f}\n")
  440. print(f"\nSaved: {csv_path}")
  442. # NPZ ensemble data (for downstream scripts and figure generation)
  443. npz_path = os.path.join(OUTPUT_DIR, 'network_ensemble_data.npz')
  444. np.savez_compressed(npz_path, **ensemble_data)
  445. print(f"Saved: {npz_path}")
  448. # ══════════════════════════════════════════════════════════════
  449. # ENTRY POINT
  450. # ══════════════════════════════════════════════════════════════
  451. if __name__ == '__main__':
  452. results, ensemble_data = run_ensemble()
  453. print_results(results)
  454. save_results(results, ensemble_data)
  455. print("\n✓ 01_network_generation.py complete.")

01_network_generation.py at commit 04d398f, under MIT · at the source

Overview

Authors: Chulwook Park1,2,3
ORCID iDs: Chulwook Park
  1. Department of Physical Education, Seoul National University, Seoul, South Korea
  2. Systemic Risk and Resilience, International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria
  3. Complexity Science and Evolution, Okinawa Institute of Science and Technology (OIST), Okinawa, Japan
Institutions: International Institute for Applied Systems Analysis (Austria); Seoul National University (South Korea); Okinawa Institute of Science and Technology Graduate University (Japan)
Journal: PLoS computational biology, volume 22, issue 7, article e1014523
Dates: received 6 March 2026; accepted 1 July 2026; published online 21 July 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.1371/journal.pcbi.1014523 · PMID 42479799 · PMCID PMC13423191 · OpenAlex W7169861276
Open access: gold, a free copy (OpenAlex)
Status: code verified
Categories: computational modeling (no new data) (modality), human (organism), computational (subfield)
Methods: Statistics, Smoothing, state filtering, decompositions, Graphs, Physiology & signal measures, Spectral & time-frequency
MeSH: Learning*, Models, Neurological*, Motor Skills*, Nerve Net*, Brain, Computational Biology, Computer Simulation, Humans (* major topic)
Topic: Motor Control and Adaptation (Cognitive Neuroscience, Neuroscience), according to OpenAlex
Funding: National Research Foundation of Korea (2020R1I1A1A01056967); Ministry of Education (MOE), Korea – BK21 FOUR Program
Citations: not cited yet (Europe PMC); 71 references in the paper

Abstract

Coordinative structures are the functional groupings of degrees of freedom that simplify motor control. Decades of research have documented them well, yet they remain mechanistically unexplained. How they emerge and why they are hierarchically organized are both unresolved questions. This study proposes scale-free network topology as the missing mechanism. Systematic simulations compared random networks, small-world networks, and scale-free networks. Scale-free organization reproduced the defining features of coordinative structures more completely than either alternative. Those features are a hub-periphery hierarchy, ordered hub-first recruitment, an abrupt onset of global coordination, and a balance between stability and flexibility. Four formal correspondences connect these features to established coordination phenomena. A coupled learning model reproduced the characteristic curve of skill acquisition, and scale-free networks reached the coordination threshold fastest. The model was then validated against published data from five independent studies spanning motor learning, bimanual coordination, brain networks, and joint coordination. It yields eleven testable predictions with explicit quantitative thresholds, together with five criteria that would disconfirm it. Network topology therefore offers a principled account of how coordinative structures form through structural constraints and experience. That account is empirically grounded and open to falsification.

Reproduced under the paper's license (CC BY), from the paper cited above.

Repositories

Its files are read in the Code ↔ Paper reader above, with 14 matches between paragraphs and lines of code.

pcw8531/Coordinative-structures-scale-free-networks

License: MIT
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Commit: 04d398f99cf12af41cbd2324fba7fa8754bf5036, 7 July 2026
Languages: Python (4)
Size: 16 files, 4 scripts
Software Heritage: not archived
Found in: “Data Availability”
Holds: README, license file, environment (requirements.txt)
Not found: CITATION.cff, tests, continuous integration, documentation
Tools: NetworkX (4 files), NumPy (4 files), SciPy (1 file)
Availability: 1 check, the latest on 27 September 2026: the link answers
  • 27 September 2026: the link answers
6 files
At the source: github.com/pcw8531/Coordinative-structures-scale-free-networks

Zenodo 20694466

License: MIT
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Size: 1 file
Software Heritage: not checked
Found in: “Data Availability”
Not found: README, license file, CITATION.cff, environment file, tests, continuous integration, documentation
Tools: NetworkX (4 files), NumPy (4 files), SciPy (1 file)
Availability: 1 check, the latest on 27 September 2026: the link answers (HTTP 200)
  • 27 September 2026: the link answers (HTTP 200)
6 files
At the source: doi.org/10.5281/zenodo.20694466

The paper's code and data availability statement is in the Data section.

Tracing map

Proposed by the machine: these links were found in the paper and verified at the source, without human review. The map will receive a Zenodo DOI once one of the paper's authors has validated it with their ORCID.

What the map holds:

  • 2 repositories of the authors' code, each at its verified commit, with its license and how the link was found in the paper;
  • 8 scripts, each with its path and the digest of its content;
  • 14 matches between paragraphs of the paper and lines of the code (method lexical-v1);
  • neither the text of the paper nor the code itself.

Its JSON (tracing-map.json) is deposited on Zenodo with its DOI once the map is validated.

Data

No dataset and no data link were found in the paper.

Data Availability

All data underlying the findings are within the paper and its Supporting information files. The complete simulation source code is openly available under an MIT license at https://github.com/pcw8531/Coordinative-structures-scale-free-networks and archived on Zenodo at https://doi.org/10.5281/zenodo.20694466.

Reproduced under the paper's license (CC BY), from the paper cited above.

Versions

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Version 1, 27 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 1 author, 8 MeSH terms, 2 funders, 47 references.

Cite

This paper

Park, C. (2026). Coordinative structures as scale-free networks: Cascade and percolation dynamics in motor learning with empirical validation. PLoS computational biology, 22(7), e1014523. https://doi.org/10.1371/journal.pcbi.1014523

BibTeX

@article{park2026coordinative,
author = {Park, Chulwook},
title = {{Coordinative structures as scale-free networks: Cascade and percolation dynamics in motor learning with empirical validation}},
journal = {PLoS computational biology},
year = {2026},
month = jul,
volume = {22},
number = {7},
pages = {e1014523},
publisher = {PLOS},
issn = {1553-734X},
doi = {10.1371/journal.pcbi.1014523},
url = {https://doi.org/10.1371/journal.pcbi.1014523},
pmid = {42479799},
pmcid = {PMC13423191}
}

RIS

TY - JOUR
AU - Park, Chulwook
TI - Coordinative structures as scale-free networks: Cascade and percolation dynamics in motor learning with empirical validation
T2 - PLoS computational biology
J2 - PLoS Comput Biol
PY - 2026
DA - 2026/07/21
VL - 22
IS - 7
SP - e1014523
SN - 1553-734X
PB - PLOS
DO - 10.1371/journal.pcbi.1014523
UR - https://doi.org/10.1371/journal.pcbi.1014523
LA - en
ER -

CSL-JSON

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The tracing map gets a citation of its own once an author has validated it and it has a DOI.

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